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Mirrors > Home > MPE Home > Th. List > submrc | Structured version Visualization version Unicode version |
Description: In a closure system which is cut off above some level, closures below that level act as normal. (Contributed by Stefan O'Rear, 9-Mar-2015.) |
Ref | Expression |
---|---|
submrc.f | mrCls |
submrc.g | mrCls |
Ref | Expression |
---|---|
submrc | Moore |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | submre 16265 | . . . 4 Moore Moore | |
2 | 1 | 3adant3 1081 | . . 3 Moore Moore |
3 | simp1 1061 | . . . 4 Moore Moore | |
4 | submrc.f | . . . 4 mrCls | |
5 | simp3 1063 | . . . . 5 Moore | |
6 | mress 16253 | . . . . . 6 Moore | |
7 | 6 | 3adant3 1081 | . . . . 5 Moore |
8 | 5, 7 | sstrd 3613 | . . . 4 Moore |
9 | 3, 4, 8 | mrcssidd 16285 | . . 3 Moore |
10 | 4 | mrccl 16271 | . . . . 5 Moore |
11 | 3, 8, 10 | syl2anc 693 | . . . 4 Moore |
12 | 4 | mrcsscl 16280 | . . . . . 6 Moore |
13 | 12 | 3com23 1271 | . . . . 5 Moore |
14 | fvex 6201 | . . . . . 6 | |
15 | 14 | elpw 4164 | . . . . 5 |
16 | 13, 15 | sylibr 224 | . . . 4 Moore |
17 | 11, 16 | elind 3798 | . . 3 Moore |
18 | submrc.g | . . . 4 mrCls | |
19 | 18 | mrcsscl 16280 | . . 3 Moore |
20 | 2, 9, 17, 19 | syl3anc 1326 | . 2 Moore |
21 | 2, 18, 5 | mrcssidd 16285 | . . 3 Moore |
22 | inss1 3833 | . . . 4 | |
23 | 18 | mrccl 16271 | . . . . 5 Moore |
24 | 2, 5, 23 | syl2anc 693 | . . . 4 Moore |
25 | 22, 24 | sseldi 3601 | . . 3 Moore |
26 | 4 | mrcsscl 16280 | . . 3 Moore |
27 | 3, 21, 25, 26 | syl3anc 1326 | . 2 Moore |
28 | 20, 27 | eqssd 3620 | 1 Moore |
Colors of variables: wff setvar class |
Syntax hints: wi 4 w3a 1037 wceq 1483 wcel 1990 cin 3573 wss 3574 cpw 4158 cfv 5888 Moorecmre 16242 mrClscmrc 16243 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-int 4476 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-mre 16246 df-mrc 16247 |
This theorem is referenced by: evlseu 19516 |
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